
At \(e=0\) , the invariance under the group of area-preserving
diffeomorphisms is promoted to full diffeomorphism invariance, and the
ambiguity (REF ) is reduced to a one-parameter ambiguity
depending only on the topology of the surface \(\Sigma \) .
This looks similar to the quantum field theory describing the weak-coupling limit of
Yang-Mills theory without defects [1]}, [2]}.
The difference is that here \(\phi \)  takes values in a different bundle and that the path
integral is taken over twisted bundles rather than ordinary
bundles. Integrating over \(\phi \)  in (REF ) at \(e=0\) 
produces a formal delta-functional \(\delta (F_A)\) , and the
remaining path integral over \(\mathsf {Bun}^\nabla _{G\downarrow D}(\Sigma )\)  therefore localises on the locus \(F_A=0\) , which by
definition is the moduli space \({M}_G^D(\Sigma )\)  of flat
\(D\) -twisted \(G\) -bundles on \(\Sigma \) . In the usual untwisted
case [1]}, the argument showing that the resulting path integral
measure induces the correct symplectic volume form on the moduli space \({M}_G(\Sigma )\simeq \mathsf {Hom}\big (\pi _1(\Sigma ),G\big )/ G \)  of flat
\(G\) -connections on \(\Sigma \)  uses a careful application of Faddeev-Popov gauge
fixing and the triviality of analytic torsion on oriented
surfaces, together with a judicious choice of \(\upsilon _1\) . It should be possible to extend these arguments to the
twisted case.
